CptS 440 / 540
Artificial Intelligence
Uncertainty Reasoning
Non-monotonic Logic
• Traditional logic is monotonic
– The set of legal conclusions grows monotonically with
the set of facts appearing in our initial database
• When humans reason, we use defeasible logic
– Almost every conclusion we draw is subject to reversal
– If we find contradicting information later, we’ll want
to retract earlier inferences
• Nonmonotonic logic, or defeasible reasoning,
allows a statement to be retracted
• Solution: Truth Maintenance
– Keep explicit information about which
facts/inferences support other inferences
– If the foundation disappears, so must the conclusion
Uncertainty
• On the other hand, the problem might not be in the
fact that T/F values can change over time but rather
that we are not certain of the T/F value
• Agents almost never have access to the whole truth
about their environment
• Agents must act in the presence of uncertainty
– Some information ascertained from facts
– Some information inferred from facts and knowledge
about environment
– Some information based on assumptions made from
experience
Environment Properties
•
•
•
•
•
•
Fully observable vs. partially observable
Deterministic vs. stochastic / strategic
Episodic vs. sequential
Static vs. dynamic
Discrete vs. continuous
Single agent vs. multiagent
Uncertainty Arises Because of Several Factors
• Incompleteness
– Many rules are incomplete because too many
conditions to be explicitly enumerated
– Many rules incomplete because some conditions
are unknown
• Incorrectness
Where Do Probabilities Come From?
• Frequency
• Subjective judgment
• Consider the probability that the sun will still
exist tomorrow.
• There are several ways to compute this
• Choice of experiment is known as the
reference class problem
Acting Under Uncertainty
• Agents must still act even if world not certain
• If not sure which of two squares have a pit and must enter one of
them to reach the gold, the agent will take a chance
• If can only act with certainty, most of the time will not act. Consider
example that agent wants to drive someone to the airport to catch a
flight, and is considering plan A90 that involves leaving home 60
minutes before the flight departs and driving at a reasonable speed.
Even though the Pullman airport is only 5 miles away, the agent will
not be able to reach a definite conclusion - it will be more like “Plan
A90 will get us to the airport in time, as long as my car doesn't break
down or run out of gas, and I don't get into an accident, and there are
no accidents on the Moscow-Pullman highway, and the plane doesn't
leave early, and there's no thunderstorms in the area, …”
• We may still use this plan if it will improve our situation, given known
information
• The performance measure here includes getting to the airport in time,
not wasting time at the airport, and/or not getting a speeding ticket.
Limitation of Deterministic Logic
• Pure logic fails for three main reasons:
• Laziness
– Too much work to list complete set of antecedents or
consequents needed to ensure an exceptionless rule,
too hard to use the enormous rules that result
• Theoretical ignorance
– Science has no complete theory for the domain
• Practical ignorance
– Even if we know all the rules, we may be uncertain
about a particular patient because all the necessary
tests have not or cannot be run
Probability
• Probabilities are numeric values between 0 and 1
(inclusive) that represent ideal certainties (not
beliefs) of statements, given assumptions about
the circumstances in which the statements apply.
• These values can be verified by testing, unlike
certainty values. They apply in highly controlled
situations.
Probability(event) = P(event) =
#instances of the event
total #instances
Example
• For example, if we roll two dice, each showing one of six possible
numbers, the number of total unique rolls is 6*6 = 36. We
distinguish the dice in some way (a first and second or left and right
die). Here is a listing of the joint possibilities for the dice:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
• The number of rolls which add up to 4 is 3 ((1,3), (2,2), (3,1)), so the
probability of rolling a total of 4 is 3/36 = 1/12.
• This does not mean 8.3% true, but 8.3% chance of it being true.
Probability Explanation
• P(event) is the probability in the absence of any additional
information
• Probability depends on evidence.
• Before looking at dice: P(sum of 4) = 1/12
• After looking at dice: P(sum of 4) = 0 or 1, depending on what we
see
• All probability statements must indicate the evidence with respect
to which the probability is being assessed.
• As new evidence is collected, probability calculations are updated.
• Before specific evidence is obtained, we refer to the prior or
unconditional probability of the event with respect to the evidence.
After the evidence is obtained, we refer to the posterior or
conditional probability.
Probability Distributions
• If we want to know the probability of a variable
that can take on multiple values, we may define a
probability distribution, or a set of probabilities
for each possible variable value.
• TemperatureToday =
{Below50, 50s, 60s, 70s, 80s, 90sAndAbove}
• P(TemperatureToday) =
{0.1, 0.1, 0.5, 0.2, 0.05, 0.05}
• Note that the sum of the probabilities for
possible values of any given variable must always
sum to 1.
Joint Probability Distribution
• Because events are rarely isolated from other events, we
may want to define a joint probability distribution, or
P(X1, X2, .., Xn).
• Each Xi is a vector of probabilities for values of variable Xi.
• The joint probability distribution is an
n-dimensional array of combinations of probabilities.
Wet
~Wet
Rain
0.6
0.4
~Rain
0.4
0.6
Inference by Enumeration
• To determine the probability of one variable (e.g., toothache), sum
the events in the joint probability distribution where it is true:
toothache
catch
~toothache
~catch
catch
~catch
cavity
.108
.012
.072
.008
~cavity
.016
.064
.144
.576
P (toothache) = .108 + .012 + .016 + .064 = 0.2
Axioms of Probability
• 0 <= P(Event) <= 1
• Disjunction, avb, P(avb) = P(a) + P(b) – P(a^b)
a
b
Axioms of Probability
• Negation, P(~a) = 1 – P(a)
a
Axioms of Probability
• Conditional probability
– Once evidence is obtained, the agent can use
conditional probabilities, P(a|b)
– P(a|b) = probability of a being true given that we
know b is true
P ( a ^ b)
– The equation P(a|b) = P(b)
holds whenever P(b)>0
• An agent who bets according to probabilities
that violate these axioms can be forced to bet
so as to lose money regardless of outcome
[deFinetti, 1931]
Axioms of Probability
• Conjunction
– Product rule
– P(a^b) = P(a)*P(b|a)
– P(a^b) = P(b)*P(a|b)
a b
• In order words, the only way a and b can both be true is if a is
true and we know b is true given a is true (thus b is also true)
Axioms of Probability
• If a and b are independent events (the truth of
a has no effect on the truth of b),
then P(a^b) = P(a) * P(b).
• “Wet” and “Raining” are not independent
events.
• “Wet” and “Joe made a joke” are pretty close
to independent events.
a b
a
b
More Than 2 Variables
• The chain rule is derived by successive
application of the product rule:
• P(X1,..,Xn) = P(X1,..,Xn-1)P(Xn|X1,..,Xn-1)
= P((X1,..,Xn-2)P(Xn-1|X1,..,Xn-2)P(Xn|X1,..,Xn-1)
=…
=  in1 P(Xi|X1,..,Xi-1)
X1
X2
X3
Law of Alternatives
• If we know that exactly one of A1, A2, ..., An are true,
then we know
P(B) = P(B|A1)P(A1) + P(B|A2)P(A2) + ... + P(B|An)P(An)
and
P(B|X) = P(B|A1,X) + ... + P(B|An,X)P(An,X)
• Example
– P(Sunday) = P(Monday) =.. = P(Saturday) = 1/7
– P(FootballToday) =
P(FootballToday|Sunday)P(Sunday) +
P(FootballToday|Monday)P(Monday) +
.. +
P(FootballToday|Saturday)P(Saturday)
= 0 + 0 + 0 + 0 + 0 + 0 + 1/7*1 = 1/7
Lunar Lander Example
• A lunar lander crashes somewhere in your town (one of the cells at
random in the grid). The crash point is uniformly random (the
probability is uniformly distributed, meaning each location has an
equal probability of being the crash point).
• D is the event that it crashes downtown.
• R is the event that it crashes in the river.
D
D
D
R
R
R
R
R
DR DR DR
R
R
R
R
R
R
DR DR DR
R
D
D
D
What is P(R)?
18/54
What is P(D)?
12/54
What is P(D^R)?
6/54
What is P(D|R)?
6/18
What is P(R|D)?
6/12
What is P(R^D)/P(D)? 6/12
Axioms of Probability
• Bayes' Rule
– Given a hypothesis (H) and evidence (E), and given that P(E) = 0,
what is P(H|E)?
• Many times rules and information are uncertain, yet we still
want to say something about the consequent; namely, the
degree to which it can be believed. A British cleric and
mathematician, Thomas Bayes, suggested an approach.
• Recall the two forms of the product rule:
– P(ab) = P(a) * P(b|a)
– P(ab) = P(b) * P(a|b)
• If we equate the two right-hand sides and divide by P(a),
we get P(b | a)  P(a | b) P(b)
P(a)
Example
• Bayes' rule is useful when we have three of the four parts
of the equation.
• In this example, a doctor knows that meningitis causes a
stiff neck in 50% of such cases. The prior probability of
having meningitis is 1/50,000 and the prior probability of
any patient having a stiff neck is 1/20.
• What is the probability that a patient has meningitis if they
have a stiff neck?
• H = "Patient has meningitis“
• E = "Patient has stiff neck"
P(H|E) =
P(E|H) * P(H)
P(E)
P(H|E) = (0.5*.00002) / .05 = .0002
Example
• I have three identical boxes labeled H1, H2, and H3
I place 1 black bead and 3 white beads into H1
I place 2 black beads and 2 white beads into H2
I place 4 black beads and no white beads into H3
• I draw a box at random, and randomly remove a bead from
that box. Given the color of the bead, what can I deduce as
to which box I drew?
• If I replace the bead, then redraw another bead at random
from the same box, how well can I predict its color before
drawing it?
H1
H2
H3
Answer
• Observation: I draw a white bead.
• P(H1|W) = P(H1)P(W|H1) / P(W)
= (1/3 * 3/4) / 5/12 = 3/12 * 12/5 = 36/60 = 3/5
• P(H2|W) = P(H2)P(W|H2) / P(W)
= (1/3 * 1/2) / 5/12 = 1/6 * 12/5 = 12/30 = 2/5
• P(H3|W) = P(H3)P(W|H3) / P(W)
= (1/3 * 0) / 5/12 = 0 * 12/5 = 0
Example
• If I replace the bead, then redraw another
bead at random from the same box, how well
can I predict its color before drawing it?
• P(H1)=3/5, P(H2) = 2/5, P(H3) = 0
• P(W) = P(W|H1)P(H1) + P(W|H2)P(H2) +
P(W|H3)P(H3)
= 3/4*3/5 + 1/2*2/5 + 0*0 = 9/20 + 4/20 = 13/20
H1
H2
H3
Monty Hall Problem
• Monty Hall Applet
• Another Monty Hall Applet
Example
•
•
We wish to know probability that John has malaria, given that he has a slightly unusual
symptom: a high fever.
We have 4 kinds of information
a)
b)
c)
d)
•
•
probability that a person has malaria regardless of symptoms (0.0001)
probability that a person has the symptom of fever given that he has malaria (0.75)
probability that a person has symptom of fever, given that he does NOT have malaria (0.14)
John has high fever
H = John has malaria
E = John has a high fever
P(H|E) =
Suppose P(H) = 0.0001, P(E|H) = 0.75, P(E|~H) = 0.14
P(E|H) * P(H)
P(E)
Example
• We wish to know probability that John has malaria, given that he has a slightly
unusual symptom: a high fever.
• We have 4 kinds of information
a)
b)
c)
d)
•
•
probability that a person has malaria regardless of symptoms
probability that a person has the symptom of fever given that he has malaria
probability that a person has symptom of fever, given that he does NOT have malaria
John has high fever
P(E|H) * P(H)
H = John has malaria
P(H|E) =
P(E)
E = John has a high fever
Suppose P(H) = 0.0001, P(E|H) = 0.75, P(E|~H) = 0.14
Then P(E) = 0.75 * 0.0001 + 0.14 * 0.9999 = 0.14006
and P(H|E) = (0.75 * 0.0001) / 0.14006 = 0.0005354
On the other hand, if John did not have a fever, his probability of having malaria would be
P(H|~E) =
P(~E|H) * P(H)
P(~E)
=
(1-0.75)(0.0001)
(1-0.14006)
= 0.000029
Which is much smaller.
Making Decision Under Uncertainty
• Consider the following plans for getting to the airport:
–
–
–
–
P(A25 gets me there on time | ...) = 0.04
P(A90 gets me there on time | ...) = 0.70
P(A120 gets me there on time | ...) = 0.95
P(A1440 gets me there on time | ...) = 0.9999
• Which action should I choose?
• Depends on my preferences for missing the flight vs.
time spent waiting, etc.
– Utility theory is used to represent and infer preferences
– Decision theory is a combination of probability theory and
utility theory
Belief Networks
• A belief network (Bayes net) represents the
dependence between variables.
• Components of a belief network graph:
• Nodes
– These represent variables
• Links
– X points to Y if X has a direct influence on Y
• Conditional probability tables
– Each node has a CPT that quantifies the effects the parents
have on the node
• The graph has no directed cycles
Example
• I'm at work, neighbor John calls to say my alarm is
ringing, but neighbor Mary doesn't call. Sometimes it's
set off by minor earthquakes. Is there a burglar?
• Variables: Burglar, Earthquake, Alarm, JohnCalls,
MaryCalls
Network topology reflects “causal” knowledge:
Example
• Suppose you are going home, and you want to know the
probability that the lights are on given the dog is barking
and the dog does not have a bowel problem. If the family is
out, often the lights are on. The dog is usually in the yard
when the family is out and when it has bowel troubles. If
the dog is in the yard, it probably barks.
• Use the variables:
f = family out
l = light on
b = bowel problem
d = dog out
h = hear bark
• There should be a graph with five nodes.
Example
• We know
– l is directly influenced by f and is independent of b,d,h given f
Add link from f to l
– d is directly influenced by f and b, independent of l and h
Add link from f to d and b to d
– h is directly influenced by d, independent of f, l, b, and d
Add link from d to h
f
b
l
d
h
Once we specify the topology (or learn it from data),
we need to specify the conditional probability table
for each node
p(f) = 0.15, 0.85
p(l|f) = 0.60, 0.40
p(d|f,b) = 0.99, 0.01
p(d|-f,b) = 0.97, 0.03
p(h|d) = 0.70, 0.30
p(b) = 0.01, 0.99
p(l|-f) = 0.05, 0.95
p(d|f,-b) = 0.90, 0.10
p(d|-f,-b) = 0.30, 0.70
p(h|-d) = 0.01, 0.99
Example
• Smart Home Example
• JavaBayes
• Other Free Bayes Network Software Packages
The Bad (and Challenging) News
• General querying of Bayes nets is NP-Complete
• The best known algorithm is exponential in the number of
variables
• Pathfinder system
–
–
–
–
–
Heckerman, 1991
Diagnostic system for lymph-node diseases
60 diseases, 100 symptoms and test rules
14,000 probabilities
8 hours to determine variables, 35 hours for topology, 40 hours
for CPTs
– Outperforms world experts in diagnosis
– Being extended to several dozen other medical domains
• LA Times article on belief networks
Netica
•
•
•
•
•
•
•
•
Nature nodes, decision nodes, utility nodes
Links
Learn values from observations
Probabilities (percentages) must sum to 100.0
Compile
Make observation
Calculate posterior probabilities
Netica Smart Home example
Utility Node
• Expected value of a variable is the sum of the
products of the variable values and their
probabilities
• E(Dice roll) = 1/6*1 + 1/6*2 + 1/6*3
+ 1/6*4 + 1/6*5 + 1/6*6 = 3.5
• Utility of an action is a numeric value indicating
the goodness of the outcome of the action
(utility can also apply to state)
• If actions have probabilistic outcomes, then
expected utility is probability of outcome * utility
of outcome, summed over all possible outcomes
Nondeterministic Games
• In backgammon, the dice rolls determine legal moves
Nondeterministic Games
Nondeterministic Game Algorithm
• Just like Minimax except also handle chance nodes
• Compute ExpectMinimaxValue of successors
– If n is terminal node, then ExpectMinimaxValue(n) =
Utility(n)
– If n is a Max node, then
ExpectMinimaxValue(n) = maxsSuccessors(n) ExpectMinimaxValue(s)
– If n is a Min node, then
ExpectMinimaxValue(n) = minsSuccessors(n) ExpectMinimaxValue(s)
– If n is a chance node, then
ExpectMinimaxValue(n) =
sSuccessors(n) P(s) * ExpectMinimaxValue(s)
Game Theory
• Decision problems in which utility of an action depends on environment
AND on actions of other agents
• Assume agents make decisions simultaneously without knowledge of
decisions of other agents
• Trading Agent Competition
Prisoner’s Dilemma
• Problem drawn from political science and game theory
•
•
•
•
•
•
•
•
Two players, each with a choice of cooperating with the other or defecting
Each receives payoff according to payoff matrix for their decision
When both cooperate, both rewarded equal, intermediate payoff (reward, R)
When one player defects, he/she receives highest payoff (temptation, T)
and other gets poor payoff (sucker, S)
When both player defect they receive intermediate penalty P
Make problem more interesting by repeating with same players, use history to
guide future decisions (iterated prisoner's dilemma)
Some strategies:
Tit For Tat:
– Cooperate on first move then do whatever opponent did on previous move, performed
best in tournament
• Golden Rule:
– Always cooperate
• Iron Rule:
– Always defect
Examples
• In the first example, the other player chooses
randomly
• Prisoner's Dilemma Applet
• Visualize Prisoner's Dilemma
Fuzzy Logic
• “Precision carries a cost”
– Boolean logic relies on sharp distinctions
– 6’ is tall, 5’ 11 ½” is not tall
• The tolerance for imprecision feeds human
capabilities
– Example, drive in city traffic
• Fuzzy logic is NOT logic that is fuzzy
– Logic that is used to describe fuzziness
Fuzzy Logic
• Fuzzy Logic is a multivalued logic that allows intermediate values to
be defined between conventional evaluations like yes/no,
true/false, black/white, etc.
• Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor of
computer science at the University of California in Berkeley.
• The concept of fuzzy sets is associated with the term ``graded
membership''.
• This has been used as a model for inexact, vague statements about
the elements of an ordinary set.
• Fuzzy logic prevalent in products:
–
–
–
–
–
Washing machines
Video cameras
Razors
Dishwasher
Subway systems
Fuzzy Sets
• In a fuzzy set the elements have a DEGREE of
existence.
• Some typically fuzzy sets are large numbers,
tall men, young children, approximately equal
to 10, mountains, etc.
Fuzzy Sets
Ordinary Sets
1
If x in A
0
If x not in A
fA(x) =
A Fuzzy Set has Fuzzy Boundaries
• A fuzzy set A of universe X is defined by
function fA(x) called the membership function
of set A
fA(x) = {0, 1}, where
•
•
•
•
•
•
fA(x) = 1 if x is totally in A;
fA(x) = 0 if x is not in A;
0 < fA(x) < 1 if x is partly in A.
fA(x) = i, where 0 <= i <= 1
If fA(x) > fA(y), then x is “more in” the set than y
If fA(x) = 1, then x in A
If fA(x) = 0, then x in A
If fA(x) = , where 0 < < 1, then x A
Degree of membership sometimes determined as a function (degree of tall
calculated as a function of height)
Fuzzy Sets
Degree of Membership
Name
Height, cm
Chris
208
Crisp
1
Fuzzy
1.00
Mark
205
1
1.00
John
198
1
0.98
Tom
181
1
0.82
David
179
0
0.78
Mike
172
0
0.24
Bob
167
0
0.15
Steven
158
0
0.06
Bill
155
0
0.01
Peter
152
0
0.00
Fuzzy Set Representation
Degree of
Membership
1.0
Crisp Sets
Short
0.8
Average
Short
Tall
Tall Men
0.6
0.4
0.2
0.0
A man who is 184 cm tall is a member
of the average men set with a degree
of membership of 0.1
150
160
170
180
190
200
210
Height, cm
Degree of
Membership
1.0
Fuzzy Sets
0.8
Short
0.6
At the same time, he is also a
member of the tall men set with a
degree of 0.4.
Tall
Average
0.4
Tall
0.2
0.0
150
160
170
180
190
200
210
Fuzzy Set Representation
• Typical functions that can be used to represent a
fuzzy set are
– Sigmoid
– Gaussian
– Linear fit (preferred because low computation cost)
 (x)
X
Fuzzy Subset A
1
0
Crisp Subset A
Fuzziness
Fuzziness
x
Linguistic Variables and Hedges
• In fuzzy expert systems, linguistic variables are used in fuzzy rules. For
example:
IF
THEN
wind
sailing
is strong
is good
IF
THEN
project_duration
completion_risk
is long
is high
IF
THEN
speed
is slow
stopping_distance is short
Linguistic Variables and Hedges
• The range of possible values of a linguistic
variable represents the universe of discourse of
that variable.
– Example, speed
– University of discourse might have range 0 .. 220 mph
– Fuzzy subsets might be very slow, slow, medium, fast,
and very fast.
• Hedges
– Modify the shape of fuzzy sets
– Adverbs such as very, somewhat, quite, more or less
and slightly.
Linguistic Variables and Hedges
Degree of
Membership
1.0
Short
0.8
Short
Tall
Average
0.6
0.4
0.2
Very Short
Very
Very
Tall
Tall
Tall
0.0
150
160
170
180
190
200
210
Height, cm
Fuzzy Set Relations
• One set A is a subset of set B if for every x, fA(x) <= fB(x)
Sets A and B are equal if for every element x, fA(x) = fB(x).
• OR / Union
– AUB is the smallest fuzzy subset of X containing both A and B,
and is defined by fA B= max(fA(x),fB(x))
U
• AND / Intersection
– The intersection A B is the largest fuzzy subset of X contained
in both A and B, and is defined by fA B(x) = min(fA(x), fB(x))
• NOT: truth(~x) = 1.0 - truth(x)
• IMPLICATION: A -> B = ~A v B, so
truth(A->B) = max(1.0 – fA(x), fB(x))
Examples
•
•
•
•
•
•
Fuzzy Logic Washing Machine
Fuzzy Logic Rice Cooker
Fuzzy Logic Barcode Scanner
Fuzzy Logic Blender
Fuzzy Logic Shampoo
Fuzzy Logic Monitor